Testing Statistical Graphics: Eye Fitting Straight Lines in the Modern Era
Cal Poly Masters Seminar
2024-07-29
Introduction to Visual Inference
When doing exploratory data analysis, how do we know if what we see is actually there?
Lineup Protocol (Buja et al. 2009)
Embed a target plot (actual data) in a lineup of null plots (randomly permuted data sets).
Introduction to Visual Inference
Linear Regression
The principle of simple linear regression is to find the line (i.e., determine its equation) which passes as close as possible to the observations, that is, the set of points.
Let’s see your drawing skills!
OR VISIT bit.ly/3BF56Zj
Did it looks something like this?
Linear Regression
The principle of simple linear regression is to find the line (i.e., determine its equation) which passes as close as possible to the observations, that is, the set of points.
Big Idea: How do statistical regression results compare to intuitive, visually fitted results?
Eye Fitting Straight Lines
Mosteller et al. (1981)
- Big Idea: Students fitted lines by eye to four sets of points.
- Method: 8.5 x 11 inch transparency with a straight line etched across the middle.
- Sample: 153 graduate students and post docs in Introductory Biostatistics.
- Experimental Design: Latin square.
- Findings: Students tended to fit the slope of the first principal component.
‘You Draw It’ Feature
(New York Times, 2015)
Readers are asked to input their own assumptions about various metrics and compare how these assumptions relate to reality.
‘You Draw It’ Task
Study Participant Prompt: Use your mouse to fill in the trend in the yellow box region.
Development & Implementation
Robinson, Howard, and VanderPlas (2023) provides details on the development and implementation of the ‘You Draw It’ method in
R.youdrawitRpackage (Dillon Murphy, Google Summer of Code 2023).
Data Generation
\(N = 30\) points \((x_i, y_i), i = 1,...N\) were generated for \(x_i \in [x_{min}, x_{max}]\).
Data were simulated based on linear model with additive errors: \[\begin{equation} y_i = \beta_0 + \beta_1 x_i + e_i \end{equation}\]
where \(e_i \sim N(0, \sigma^2).\)
Parameters \(\beta_0\) and \(\beta_1\) were selected to reflect the four data sets used in Mosteller et al. (1981).
Study Design
Experiment conducted and distributed through an RShiny application found here.
Model Data
For each participant, the final data set used for analysis contains: + \(x_{ijk}\), \(y_{ijk,drawn}\), \(\hat y_{ijk,OLS}\), \(\hat y_{ijk,PCA}\)
for + parameter choice \(i = 1,2,3,4\), + participant j = \(1,...N_{participant}\) + \(x_{ijk}\) value corresponding to increment \(k = 1, ...,4 x_{max} + 1\).
Vertical residuals between the drawn and fitted values were calculated as: + \(e_{ijk,OLS} = y_{ijk,drawn} - \hat y_{ijk,OLS}\) + \(e_{ijk,PCA} = y_{ijk,drawn} - \hat y_{ijk,PCA}\).
Linear Trend Constraint
Smoothing Spline Trend
